Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Empirical Watershed Wavelet

Basile Hurat, Zariluz Alvarado, Jerome Gilles

TL;DR

This paper provides theoretical results that permits us to build 2D empirical wavelet filters based on an arbitrary partitioning of the frequency domain and proposes an algorithm to detect such partitioning from an image spectrum.

Abstract

The empirical wavelet transform is an adaptive multiresolution analysis tool based on the idea of building filters on a data-driven partition of the Fourier domain. However, existing 2D extensions are constrained by the shape of the detected partitioning. In this paper, we provide theoretical results that permits us to build 2D empirical wavelet filters based on an arbitrary partitioning of the frequency domain. We also propose an algorithm to detect such partitioning from an image spectrum by combining a scale-space representation to estimate the position of dominant harmonic modes and a watershed transform to find the boundaries of the different supports making the expected partition. This whole process allows us to define the empirical watershed wavelet transform. We illustrate the effectiveness and the advantages of such adaptive transform, first visually on toy images, and next on both unsupervised texture segmentation and image deconvolution applications.

The Empirical Watershed Wavelet

TL;DR

This paper provides theoretical results that permits us to build 2D empirical wavelet filters based on an arbitrary partitioning of the frequency domain and proposes an algorithm to detect such partitioning from an image spectrum.

Abstract

The empirical wavelet transform is an adaptive multiresolution analysis tool based on the idea of building filters on a data-driven partition of the Fourier domain. However, existing 2D extensions are constrained by the shape of the detected partitioning. In this paper, we provide theoretical results that permits us to build 2D empirical wavelet filters based on an arbitrary partitioning of the frequency domain. We also propose an algorithm to detect such partitioning from an image spectrum by combining a scale-space representation to estimate the position of dominant harmonic modes and a watershed transform to find the boundaries of the different supports making the expected partition. This whole process allows us to define the empirical watershed wavelet transform. We illustrate the effectiveness and the advantages of such adaptive transform, first visually on toy images, and next on both unsupervised texture segmentation and image deconvolution applications.

Paper Structure

This paper contains 22 sections, 1 theorem, 21 equations, 9 figures, 8 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. The Empirical Wavelet Transform
  3. 1D Empirical Wavelet Transforms
  4. 2D Empirical Wavelet Transforms
  5. Scale-Space Representations
  6. A New 2D Empirical Wavelet Transform
  7. 2D empirical wavelets of arbitrary supports
  8. On the detection of partitions of arbitrary geometry
  9. Scale-space localization of harmonic modes
  10. Watershed Transforms
  11. The Empirical Watershed Wavelet Transform
  12. Experiments
  13. Harmonic mode decomposition
  14. Performance on Unsupervised Texture Segmentation
  15. Influence of the EWWT parameters
  16. ...and 7 more sections

Key Result

Proposition 1

Denoting $\varphi_{n,i,j}=\varphi_n(\cdot-i,\cdot-j)$, the set $\{\varphi_{n,i,j}\}$ forms a frame.

Figures (9)

  • Figure S1: Construction of 1D Empirical Wavelet: given a set of boundaries $\{\omega_i$}, we define $\widehat{\psi}_n$ as a band-pass filter between $\omega_n$ and $\omega_{n+1}$ with transition regions of width $2\tau_n$ and $2\tau_{n+1}$.
  • Figure S2: Examples of 2D EWT Implementations: In order from left to right, we see the partitioning of the Fourier spectrum of the (a) Tensor, (b) Littlewood-Paley, and (c) Curvelet (Option 1) empirical wavelet transforms.
  • Figure S4: Tracking of Maxima through Scale-Space (the vertical axis corresponds to the scale $s$): (a) shows all maxima through scale-space. (b) shows the remaining persistent maxima, i.e of lifespan larger than the threshold defined by Otsu's method.
  • Figure S5: Step by step construction of empirical watershed wavelets: (a) Original image, (b) Magnitude spectrum of image, (c) Detected persistent maxima using method from Section \ref{['sec:3:modedetection']}, (d) detected partitioning of the spectrum using method from Section \ref{['sec:3:watershed']}, (e) Paired regions if image is real, (f) Distance transform of central region, (g) Empirical watershed wavelet of central region, (h) empirical watershed wavelet coefficient for central region.
  • Figure S6: (a) An example image, (b) its magnitude Fourier transform, (c) the detected partition using the EWWT.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Proposition 1